The limited dependence between the additive and the multiplicative structure of fields is in the background of a number of explicit constructions of various types of pseudorandom objects. In this direction we study the size of the intersection of (the additive) translates of fibers of the (multiplicative) norm function over finite fields. Besides extending earlier upper bounds, our main focus here is on obtaining lower bounds.

From our results we conclude several consequences in extremal combinatorics. Our motivation is the projective norm graph $\text {NG}(q,t)$ and its small subgraph statistics. $\text {NG}(q,t)$ provides a tight construction for the Turán number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$; in particular, it does not contain $K_{t, (t-1)!+1}$. Yet, for $t\geq 4$ it is not even known whether $\text {NG}(q,t)$ contains $K_{t,t}$. The determination of the largest integer $s(t)$, such that $\text {NG}(q,t)$ contains $K_{t,s(t)}$ for all large enough prime powers q is an important open question with far-reaching consequences, and the best known bounds, $t-1\leq s(t) \leq (t-1)!$, are very far apart. In this paper we settle the first open case and establish $s(4)=6$. Along the way we also count subgraphs of $\text {NG}(q,t)$ isomorphic to H, for any fixed $3$-degenerate graph H, and find that projective norm graphs are quasirandom with respect to these parameters. These results go beyond the consequences of the Expander Mixing Lemma and also imply extensions of the work of Alon and Shikhelman on generalized Turán numbers.

Finally, we also give an elementary proof of the $K_{4,s}$-freeness of $\text {NG}(q,t)$ for $s= 6(\sum _{i=0}^{t-4} q^i) +1$. This was known before for $t=4$ only, via a less direct algebro-geometric argument.

Let ${\mathbb F}$ be an algebraically closed field and $G$ be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups $H<G$ and ${\mathbb F} G$-modules $L$ such that the restriction $L{{\downarrow }}_H$ is irreducible. This problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $G$ is the Schur’s double cover of an alternating or symmetric group.

Let G be a product of unitary groups and let $(M,\omega )$ be a compact symplectic manifold with Hamiltonian G-action. We prove an equivariant formality result for any complex-oriented cohomology theory $\mathbb {E}^*$ (in particular, integral cohomology). This generalizes the celebrated result of Atiyah–Bott–Kirwan for rational cohomology from the 1980s. The proof does not use classical ideas but instead relies on a recent cohomological splitting result of Abouzaid–McLean–Smith [AMS21] for Hamiltonian fibrations over $\mathbb {CP}^1.$ Moreover, we establish analogues of the ‘localization’ and ‘injectivity to fixed points’ theorems for certain cohomology theories studied by Hopkins–Kuhn–Ravenel in [HKR00]. As an application of these results, we establish a Goresky–Kottwitz–MacPherson theorem with Morava K-theory coefficients for Hamiltonian T-manifolds.

We call a family of $s$ sets $\{F_1, \ldots , F_s\}$ a sunflower with $s$ petals if, for any distinct $i, j \in [s]$, one has $F_i \cap F_j = \cap _{u = 1}^s F_u$. The set $C = \cap _{u = 1}^s F_u$ is called the core of the sunflower. It is a classical result of Erdős and Rado that there is a function $\phi (s,k)$ such that any family of $k$-element sets contains a sunflower with $s$ petals. In 1977, Duke and Erdős asked for the size of the largest family $\mathcal {F}\subset {[n]\choose k}$ that contains no sunflower with $s$ petals and core of size $t-1$. In 1987, Frankl and Füredi asymptotically solved this problem for $k\geqslant 2t+1$ and $n>n_0(s,k)$. Their paper is one of the pinnacles of the so-called Delta-system method.

In this paper, we extend the result of Frankl and Füredi to a much broader range of parameters: $n>poly(s,t) k$, when additionally $k>poly(s,t)\log n$ or $n>n_0(s,t)$. We also extend this result to other domains, such as $[n]^k$ and ${n\choose k/w}^w$ and obtain even stronger and more general results for forbidden sunflowers with core at most $t-1$ (including results for families of permutations and subfamilies of the $k$-th layer in a simplicial complex).

The methods of the paper, among other things, combine the spread approximation approach, introduced by Zakharov and the first author, with the Delta-system approach of Frankl and Füredi and the hypercontractivity for global functions approach, developed by Keller, Lifshitz, and coauthors. Previous works in extremal set theory relied on at most one of these approaches. Creating such a unified view was one of the goals for the paper.

We establish the full asymptotic stability of solitary waves for the focusing cubic Schrödinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the symmetries of the problem, namely the invariance under scaling and phase shifts. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the nonselfadjoint linearized matrix Schrödinger operator around the solitary waves. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

We develop the deformation theory of primitive Enriques varieties, which are defined as quasi-étale quotients of primitive symplectic varieties by nonsymplectic group actions. In particular, we establish a local Torelli theorem for primitive Enriques varieties. As applications thereof, we describe the behavior of certain primitive Enriques varieties under locally trivial deformations.

Let G be a totally disconnected locally compact (tdlc) group. The contraction group $\mathrm {con}(g)$ of an element $g\in G$ is the set of all $h\in G$ such that $g^n h g^{-n} \to 1_G$ as $n \to \infty $. The nub of g can then be characterized as the intersection $\mathrm {nub}(g)$ of the closures of $\mathrm {con}(g)$ and $\mathrm {con}(g^{-1})$.

Contraction groups and nubs provide important tools in the study of the structure of tdlc groups, as already evidenced in the work of G. Willis. It is known that $\mathrm {nub}(g) = \{1\}$ if and only if $\mathrm {con}(g)$ is closed. In general, contraction groups are not closed and computing the nub is typically a challenging problem.

Maximal Kac–Moody groups over finite fields form a prominent family of nondiscrete compactly generated simple tdlc groups. In this paper we give a complete description of the nub of any element in these groups.

For a morphism $f : X \to Y$ of schemes, we give a tropical criterion for which points of Y (valued in a field, discrete valuation ring, number ring, or Dedekind domain) lift to X. Our criterion extends the firmaments of Abramovich [Abr09] to a wide range of morphisms, even logarithmic stable maps.

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of q-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a q-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most nontrivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak {gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role in the description of this embedding.

We prove that the Grothendieck ring of the monoidal category of finitely generated graded projective modules for the nil-Brauer category is isomorphic to an integral form of the split iquantum group of rank one. Under this isomorphism, the indecomposable graded projective modules correspond to the icanonical basis. We also derive character formulae for irreducible graded modules and deduce various branching rules.

For a finite group A with normal subgroup G, a subgroup U of G is an A-prime-power-covering subgroup if U meets every A-conjugacy-class of elements of G of prime-power order. It is conjectured that $|G:U|$ is bounded by some function of $|A:G|$. We prove the conjecture in the case that the action of G on the set of right cosets of U in G is innately transitive. This includes the special case where U is a maximal subgroup of G. It turns out that each prime-power-covering subgroup U for which this G-action is innately transitive must be a maximal subgroup of G. We derive the following number-theoretic consequence: if K is a finite Galois extension of a global field k and L is a finite innate extension of K such that K and L are weakly Kronecker equivalent over k, then the extension $L/K$ must be atomic of degree bounded in terms of $[K:k]$. Our proofs use both a recently proved bound on the order of a finite nonabelian simple group in terms of the number of its classes of elements of prime-power order (depending on the Finite Simple Group Classification) and also new Galois-theoretic concepts for field extensions.

We propose a definition of expander representations of quivers, generalizing dimension (or linear-algebra) expanders, as a qualitative refinement of slope stability. We prove existence of uniform expander representations for any wild quiver over an algebraically closed base field, using the concept of general subrepresentations and spectral properties of Cartan matrices.

We classify the complementary vectors of doubly Cohen–Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid. Our technique works more generally for certain level quotients of Stanley–Reisner rings, giving new bounds on the face numbers of Buchsbaum* complexes.

We study the sublevel sets of the Willmore energy on the space of smoothly immersed $ 2 $-spheres in Euclidean $ 3 $-space. We show that the subset of immersions with energy at most $ 12\pi $ consists of four regular homotopy classes. Moreover, we show that, in certain regular homotopy classes, all singularities of the Willmore flow are avoidable – that is, the initial surface admits a regular homotopy to a round sphere whose Willmore energy does not exceed that of the initial surface. This yields a classification of initial surfaces with energy at most $ 12\pi $ that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li–Yau inequality at $ 12\pi $ for a large class of immersed spheres without triple points. To prove these results, we glue together different instances of the Willmore flow and employ an invariant for triple-point-free immersed spheres.

In this paper, we prove that a compact Kähler manifold X with semi-positive holomorphic sectional curvature admits a locally trivial fibration $\phi \colon X \to Y$, where the fiber F is a rationally connected projective manifold and the base Y is a finite étale quotient of a (compact complex) torus. This result extends the structure theorem, previously established for projective manifolds, to compact Kähler manifolds. A key part of the proof involves analyzing the foliation generated by truly flat tangent vectors and showing the abelianness of the topological fundamental group $\pi _{1}(X)$, with a focus on varieties of special type.

Tight $2 s$-designs are the $2 s$-$(v, k, \lambda )$ designs whose sizes achieve the Fisher type lower bound ${v \choose s}$. Symmetric $2$-designs, the Witt $4$-$(23, 7, 1)$ design and the Witt $4$-$(23, 16, 52)$ design are tight designs. It has been widely conjectured since the 1970s that there are no other nontrivial tight designs. In this paper, we give a proof of this conjecture. In the proof, an upper bound $v \ll s$ is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound $v \gg s (\ln s)^2$ is shown by using estimates on prime gaps.

We investigate local entropy theory, particularly the property of having completely positive entropy (CPE), from a descriptive set-theoretic point of view. We aim to determine the descriptive complexity of different families of dynamical systems with CPE. For a large class of compact metrizable spaces (including manifolds and topological graphs), we show that the family of dynamical systems with CPE is complete coanalytic and hence not Borel. When we restrict our attention to dynamical systems with additional properties, we obtain contrasting behavior. Specifically, CPE and uniformly positive entropy (UPE) – the latter being a Borel property – coincide for mixing maps on topological graphs. However, the family of mixing maps with CPE on Cantor spaces is coanalytic and not Borel. For dynamical systems with the shadowing property, CPE and UPE coincide regardless of the phase space.

We construct irrational irreducible components of the Hilbert scheme of points of affine space $\mathbb {A}^n$ for $n \geq 12$. We start with irrational components of the Hilbert scheme of curves in $\mathbb {CP}^3$ and use methods developed by Jelisiejew to relate these to irreducible components of the Hilbert schemes of points of $\mathbb {A}^n$. The result solves Problem XX of the Hilbert scheme of points problem list [J24].

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset V of affine or projective space. The bounds depend only on $\mathrm {card}(\mathcal P)$ and the degrees of the polynomials in $\mathcal P$, together with $\deg (V)$ and $\dim (V)$, and not on the dimension of the space in which V is embedded. This feature is particularly useful when V has small intrinsic dimension but is presented in a very high-dimensional ambient space.

We describe several applications. First, we extend existing results on bounding the $\varepsilon $-entropy of real algebraic varieties. Second, we derive lower bounds (in terms of the number of connected components) for membership testing in semi-algebraic sets in the algebraic computation tree model. Finally, motivated by quantum complexity theory, we introduce additive and multiplicative notions of relative rank in finite-dimensional vector spaces and algebras with respect to a fixed algebraic subset, generalizing the classical notion of tensor rank. We prove a general lower bound on the maximum relative rank of finite subsets with respect to algebraic sets of bounded degree and dimension that is again independent of the ambient dimension. As an illustration, we obtain a quantum analog of Shannon’s classical lower bound. We prove that if $\Delta _n$ is an algebraic family of allowed gates with

$$ \begin{align*}\dim \Delta_n= \mathrm{poly}(n), \qquad \log\deg\Delta_n= \text{ poly}(n),\end{align*} $$

then some Boolean functions require $2^n/\text {poly}(n)$ gates from $\Delta _n$, even when individual gates in $\Delta _n$ may be highly nonlocal.

Let k be a field endowed with a complex embedding, and let$\mathcal {G}_{\mathrm {mot}}$ be the associated motivic Galois group. The goal of this paper is to show that$\mathcal {G}_{\mathrm {mot}}$ admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that$\mathcal {G}_{\mathrm {mot}}$ coincides with the automorphism group of the functor$X\mapsto \pi _1^{\mathrm {geo}}(X)$ sending a k-variety X to the geometric completion of its topological fundamental group. This can be considered as a motivic version of the Ihara–Matsumoto–Oda Conjecture, greatly generalised and proven by Pop. In a sequel to this paper, we will develop the parallel story for the$\ell $-adic realisation. For this reason, many of the intermediate results are developed at a greater level of generality than strictly necessary for the main results proven here.